1. Field of the Invention:
The present invention relates generally to restoration of a band-limited signal by signal processing techniques. More particularly, it relates to a system and method for such signal restoration that utilizes inequality constraints in order to restore the signals more accurately.
2. Description of the Prior Art:
Signal interpolation and extrapolation arise in many signal processing applications. The basic problem is to construct unknown signal samples from known samples, using the fact that the signal is band-limited, plus perhaps other information. Example applications include recovering from drop-outs in a digital audio tape, signal reconstruction after nonlinear removal of impulsive noise, and signal inference from incomplete measurements.
A sizeable literature exists on the topics of interpolation and extrapolation. When the known samples of the signal are uniform and sufficiently dense in time that the sample spacing is less than half the period of the highest frequency in the signal, band-limited interpolation provides a unique reconstruction for all time. Band-limited interpolation can be seen as replacing the known samples by a superposition of sinc functions each having spectral width equal to the signal bandwidth.
When the Nyquist sampling theorem is violated, it becomes necessary to use extrapolation techniques. Because every finite-bandwidth signal is analytic, a band-limited signal can be extrapolated for all time from any open set. The extrapolation is simply a Taylor series expansion about any point in the set. However, incompletely sampled signals cannot be so extrapolated. Extrapolation over missing, nonzero samples requires more information than the signal band limits.
A large number of extrapolation techniques construct the unique minimum norm signal which interpolates the known samples and satisfies the band-limited condition. Usually the L.sub.2 norm is minimized, yielding a minimum energy extrapolation. Minimum-norm extrapolation of lowpass signals provides a reconstruction very similar to that obtained using band-limited interpolation where zeros have been inserted for the missing samples; i.e., the extrapolation tends to go to zero after an interval approximately equal to the reciprocal of the assumed signal bandwidth. This seems reasonable when the extrapolation is viewed as a superposition of sinc functions corresponding to the signal bandwidth, along with a second-order correction that adjusts the sinc function heights so as to precisely match the known samples.
Some extrapolation techniques have been devised to make use of other a priori knowledge besides bandwidth. For example, weighting functions can be introduced in both the time and frequency domains to alter the relative strength of norm minimization versus time and frequency. Linear prediction has also been used to extrapolate the signal on the basis of correlation information measured using the available samples. In general, any known properties of the signal can help to improve the quality of its restoration from an incomplete set of samples.